Balancing Gaussian vectors in high dimension

TL;DR

This paper investigates the discrepancy of high-dimensional Gaussian matrices, providing sharp probabilistic bounds and developing an efficient algorithm for low-discrepancy matrix balancing in high dimensions.

Contribution

It offers the first sharp probabilistic bounds for Gaussian discrepancy in high dimensions and introduces a polynomial-time algorithm for low discrepancy in large random matrices.

Findings

Discrepancy of Gaussian matrices is $ heta(\sqrt{n} 2^{-n/m})$ with high probability.

An efficient randomized algorithm achieves discrepancy $e^{- ext{polylog}(n)/m}$ for matrices with i.i.d. Lipschitz continuous entries.

The algorithm matches known guarantees in one dimension and improves bounds in higher dimensions.

Abstract

Motivated by problems in controlled experiments, we study the discrepancy of random matrices with continuous entries where the number of columns $n$ is much larger than the number of rows $m$. Our first result shows that if $\omega(1) = m = o(n)$, a matrix with i.i.d. standard Gaussian entries has discrepancy $\Theta(\sqrt{n} \, 2^{-n/m})$ with high probability. This provides sharp guarantees for Gaussian discrepancy in a regime that had not been considered before in the existing literature. Our results also apply to a more general family of random matrices with continuous i.i.d entries, assuming that $m = O(n/\log{n})$. The proof is non-constructive and is an application of the second moment method. Our second result is algorithmic and applies to random matrices whose entries are i.i.d. and have a Lipschitz density. We present a randomized polynomial-time algorithm that achieves discrepancy $e^{-\Omega(\log^2(n)/m)}$ with high probability, provided that $m = O(\sqrt{\log{n}})$. In the one-dimensional case, this matches the best known algorithmic guarantees due to Karmarkar--Karp. For higher dimensions $2 \leq m = O(\sqrt{\log{n}})$, this establishes the first efficient algorithm achieving discrepancy smaller than $O( \sqrt{m} )$.